When performing machine processing of the upper surface of a workpiece, a clamp device of the type described above (i.e. so called hole clamp) is employed, since it is not possible to employ a clamp device of the type in which pressing force is exerted by a pressing member on an end portion of the workpiece from above. With such a clamp device, a seating surface on which the workpiece as an object of clamping is seated is formed on a main body member, and, when loading the workpiece, the workpiece is mounted on the seating surface and is supported thereby, and then the workpiece is fixed on the seating surface by inserting, into an aperture in the workpiece, a grip member and a tapered shaft portion of a clamp rod inserted through the grip member, by dilating grip claws with the tapered shaft portion so that they engage (i.e. grip) the inner circumferential surface of the aperture, and by then again pulling this grip member towards the seating surface. Clamp devices of this type are described in Patent Documents #1 and #2.
With the clamp devices described in these Patent Documents #1 and #2, the grip member and the tapered shaft portion are built so as to project for a predetermined short distance from the central portion of the main body member of the clamp device, and the diameter of the main body member is four times or more the diameter of the grip member.    Patent Document #1: Japanese Patent No. 3,550,010.    Patent Document #2: German Patent. No. 4,020,981.